Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Inferring observation time from a Brownian motion This might be a bit lengthy question. So let me proceed in steps.
General description: I have some observations, based on which I want to infer their occurring time.
Specific setting: Let $W(t)$ be a Brownian motion.
I have an observation $X$ defined as
$$X = BW(\delta)... | The probability $$\mathbb{P}(B = 1 \mid X) = \frac{1}{1 + \frac{1}{\sqrt{2}}e^{X^2/(4\delta)}}$$ only depends on $X$ and $\delta$ through the term $X^2/(4\delta)$. Now let's think about the cases when $B = 1$ or $B = 0$.
When $B = 1$ then $X \sim N(0, \delta)$, so $X/\sqrt{\delta} \sim N(0, 1)$. Therefore $X^2/\delta \... | {
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"url": "https://math.stackexchange.com/questions/3054629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proof about injection I have to proof that the function $f : X \rightarrow Y$ is an injection if and only if $\forall T \subseteq X$, $f(X\setminus T) \subseteq Y \setminus f(T)$.
I'm having some difficulties. First (1) I proof that if $f$ is an injection then $\forall T \subseteq X$, $f(X\setminus T) \subseteq Y \setm... | So we want to show $f$ is injective iff
$$\forall T \subseteq X: f[X\setminus T]\subseteq Y \setminus f[T]\tag{*}$$
So let $f$ be injective, $T \subseteq X$ and let $y \in f[X\setminus T]$, i.e.
$y=f(x)$ with $x \notin T$. By definition of the function, $y \in Y$, but I claim that also $y \notin f[T]$, because otherwis... | {
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"source": "stackexchange",
"question_score": "1",
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A bag contains 2 red, 3 green and 5 blue balls A ball is drawn, observed and put again in the bag. Find the probability of getting all colours different.
My teacher's solution:
2/10 × 3/10 × 5/10 × 3! × 2
I don't get why that last 2 was necessary
Edit: balls are drawn 3 times
| The last $2$ should not be there. The first three terms give the chance of getting specifically red,green,blue. The $3!$ is the number of orders of colors and you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3054913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof: For any integer $n$ with $n \ge 1$, the number of permutations of a set with $n$ elements is $n!$. I am trying to use mathematical induction to prove the following theorem:
For any integer $n$ with $n \ge 1$, the number of permutations of a set with $n$ elements is $n!$.
Proof
Let $P(n)$ be the above statemen... | When we are given $m+1$ items, we separate a special item from the other $m$ items. We sort the $m$ items and fix their order, that give us $m!$ options.
Now we have $m+1$ positions to choose to place our special items and none of them repeats.
Hence we have $m! \cdot (m+1)= (m+1)!$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Roots Across the Complex Numbers Why is it the case that an even root (square root, quartic, etc) can be positive or negative across the complex numbers, but is limited to postive in the reals? Is there a good mathematical reason for this, or is it simply notation?
| Essentially, the reason is that complex multiplication, and by extension exponentiation, are far more complicated than in the reals. There is a rotational aspect, and for an $n$th root, there are $n$ answers. It would be silly to declare that only one of these is the "actual" answer, whereas in the reals it makes muc... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Let $G$ be transitive on $S$. Show that the action is primitive if and only if every $\operatorname{Stab}_G(a), a\in S$, is a maximal subgroup of $G$. I am self-studying "Classical Groups and Geometric Algebra" by Larry C. Grove. This is the 2nd question of the exercises of the 0th Chapter.
Let $G$ be transitive on $S... | ($\implies$)
Let $H$ is a proper subgroup of $G$ containing $Stab_G(a)$ and $B=\{ha|h\in H\}$ is a subset of $S$.
Assume that $xB\cap B \neq \emptyset$ for some $x \in G$. There exist $h,h'\in H$ such that $xha=h'a$. Then, $h'^{-1}xha=a\implies h'^{-1}xh\in Stab_G(a)\implies h'^{-1}xh\in H\implies h'(h'^{-1}xh)h^{-1}=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of compositions of $n$ such that each term is less than equal to $k.$ Let $n$ be an integer $\geq 1.$ Then a partition of $n$ is a sequence of positive integers (greater than equal to $1$) such that their sum equals $n.$ So for instance if $n=4$ then
$$[[4], [1, 3], [1, 1, 2], [1, 1, 1, 1], [1, 2, 1], [2, 2], [... | First a note about terminology:
- a Partition of a positive integer $n$ is a non-decreasing sequence of positive integers summing to $n$;
- a Composition of a positive integer $n$ is an unordered sequence of positive integers summing to $n$.
That premised, you are speaking of the number of Compositions of $n$, whose ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3055314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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find the value of normal distribution with a 3 decimal places Z score using table In a past exam of my stats class, the question requires finding the value of a normal CDF corresponding to a Z score of 3 DP. However, we are only given a table that is accurate to 2 DP. We are not allowed to use calculator. What should I... | Personally, I would just round to $1.29$. However, if you really want to guess $\text{normalcdf}(1.293)$, I would just do a weighted sum.
First, write $1.293$ in terms of $1.29$ and $1.30$, since those are the two-decimal z-scores closest to it:
$$1.293=0.7\cdot 1.29+0.3\cdot 1.30$$
Then, pretend $\text{normalcdf}$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3055420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do I keep getting this incorrect solution when trying to find all the real solutions for $\sqrt{2x-3}\ +x=3$. The problem is to find all real solutions (if any exists) for $\sqrt{2x-3}\ +x=3$.
Now, my textbook says the answer is {2}, however, I keep getting {2, 6}. I've tried multiple approaches, but here is one of... | Because squaring both sides of an equation always introduces the “risk” of an extraneous solution.
As a very simple example, notice the following two equations:
$$x = \sqrt 4 \iff x = +2$$
$$x^2 = 4 \iff \vert x\vert = 2 \iff x = \pm 2$$
The first equation has only one solution: $+\sqrt 4$. The second, however, has two... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3055574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Probability of getting 6 heads in a row from 200 flips and intuition about this high value A few days ago I had an argument with a friend about this question :
What is the probability of getting 6 heads in a row from 200 flips ?
I argued it is high probability (significantly bigger than a half) while he argued it is ... | Here is how to calculate the exact answer. Consider a Markov chain $X_0,X_1,\ldots,X_{200}$, taking integer values in the range $0\le X_n\le 6$, with
transition matrix (with row and column indices in the range $0\le i,j\le6$) $$M=\pmatrix{\frac12&\frac12&0&0&0&0&0&\\
\frac12&0&\frac12&0&0&0&0&\\
\frac12&0&0&\frac12&0&... | {
"language": "en",
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How to solve this Diophantine equation? Can anyone say how one can find solutions to the Diophantine equation $$x^3+y^4=z^2$$ in General? Only a few triples of numbers have been found, and most likely this equation has infinitely many solutions.
Examples of triples: $(6,5,29),(2,1,3),(9,6,45)$...
| "OP" enquired about integer coefficent's for the parametric
solution for the equation $(x^2+y^4=z^2)$. "OP" just needs
to substitute $k=(m/n)$ in the parametrization & the resulting
solution after removing common factors is given below.
$x=6(u^3)(v^2)$
$y=(u^2)(v)(10m-27n)$
$z=(u^4)(v^2)(116m^2-540mn+621n^2)$
And $u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3055812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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I cannot calculate $\tan^{-1}(1+i)$ I use the formula below for inverse tangent function:
$$\tan^{-1}(z)= \frac {i} 2 \log \biggr( \frac {i+z}{i-z} \biggr)$$
I have written
$$\tan^{-1}(1+i)= \frac {i} 2 \log \biggr( \frac {i+1+i}{i-1-i} \biggr)= \frac {i} 2 \log (-2i-1)$$
The answer is $n\pi i , n\in \mathbb Z$ but I... | The number $-1-2i$ is in the third quadrant, so its argument, taken in $[0,2\pi)$, is
$$
\theta=\pi+\arctan\frac{-2}{-1}=\pi+\arctan2
$$
(it would be wrong to say that the argument is $\arctan2$). Thus
$$
-1-2i=\sqrt{5}e^{i\theta}
$$
and its logarithms are
$$
\frac{1}{2}\log5+i\theta+2k\pi i
$$
and therefore
$$
\frac{i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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In how many ways can I write $0$ as a sum of $n\; 0s, 1s \;\text{and}\; -1s?$ In how many ways can I write $0$ as a sum of $n\; 0s, 1s \;\text{or}\; -1s?$ (Taking the order into account).
I suspect there is no closed formula to express the result, but I'd like someone to confirm it, or deny it.
Edit:
e.g., if $n=3$
$$... | The number of $0s, 1s$ and $-1s$ possible for a particular $n$ can be seen by the number of solutions to:
$$2p+q=n| p,q \in \Bbb Z^+$$
This can be done in $\frac{n+1}{2}$ ways for odd $n$ and $\frac{n+2}{2}$ ways for even $n$
You'll just need to account for positioning after this.
| {
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"url": "https://math.stackexchange.com/questions/3056013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Meaning of "$\Leftrightarrow$" I just read (at the beginning of this Wikipedia site) that "$A \Leftrightarrow B$" means "A can be replaced in a logical proof with B". Is this interpretation of the symbol true?
(I have so far understood the symbol for equivalence ($\Leftrightarrow$) to mean that, given $A \Leftrightarro... | $$\iff \text { means *if and only if* }$$
Essentially we have that:
$$A\iff B \to (A \implies B) \text { AND } (B\implies A)$$
"A if and only if B is the same as A if B and B if A"
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3056153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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For which $a$ does the equation $a^x=x+2$ have two solutions? I need to find values of $a$ for the following equation to have two real solutions.
$$a^x=x+2$$
*
*$(1,\infty)$
*$(0,1)$
*$1/e,e$
*$(1/(e^e), e^e)$
*$(e^{1/e}, \infty)$
This is how I solved this exercise, but I don't understand some things.
I would l... | Your attempt is wrong, sorry: you cannot just use particular cases. And the case $a=-e$ is impossible, because $a^x$ is only defined for $a>0$. The answer should be in terms of $a$, and using a single value is not enough.
Consider the function $f(x)=a^x-x-2$. Then
$$
f'(x)=a^x\log a-1
$$
(with $\log$ being the natural ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Question related to Darboux's theorem Darboux's theorem says that $f'$ has intermediate value property. More precisely,
( Darboux's theorem) If $f$ is differentiable on $[a,b]$, and if r is any number for which $f'(a)<r<f'(b)$ then $\exists$ c in (a,b) such that $f'(r)=c$.
Thus Darboux's theorem implies that $f'$ can... | No, there can't be.
If $f'(x)\to +\infty$ for $x\to d$, then there will be $\delta$ such that $f'(x) > f'(d)+2$ on $(d, d+\delta)$. And this contradicts Darboux's theorem on $[d,d+\delta/2]$ -- there won't be any place with $f'(x) = f'(d)+1$.
Be a bit careful about the precise meaning of "oscillates" in your conclusion... | {
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Commutative local ring with $10$ ideals Let $R$ be a commutative ring with unity with exactly $10$ ideals (including $\{0\}$ and $R$ ) . Then is it true that $R$ is a Principal Ideal Ring ?
My Work: I know that any commutative ring with $5$ or less ideals is a PIR. Indeed, suppose $R$ has $5$ or less ideals. Then $R$ i... | I count exactly ten ideals in $R=\mathbb{F}_5[x,y]/(x^2,y^2)$. Namely
$$0,(xy),(x),(x+y),(x+2y),(x+3y),(x+4y),(y),(x,y),R.$$
And $(x,y)$ is not principal, so $R$ is not a PIR.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find the $x$ at which the local maxima of a function ocours.
The function $f(x) = \int\limits_{-1}^{x}t(e^t-1)(t-1)(t-2)^3(t-3)^5 dt$ has a local maxima at $x=?$
First, I differentiated $f(x)$ and found its roots.
That came out to be $x = 0,1,2,3$. Now, one of those numbers, when plugged into $f(x)$, must give the la... | HINT:
The sign of $$f^{'}(x)=x(e^x-1)(x-1)(x-2)^3(x-3)^5$$ is $-,+,-,+$ respectively on the intervals $(-1,1),(1,2),(2,3),(3,\infty).$ From this you can finish without further calculation.
| {
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"timestamp": "2023-03-29T00:00:00",
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How is it possible that $\textrm{HK}=\textrm{G}$? For the following problem:
If $\textrm{H}$ and $\textrm{K}$ are distinct subgroups of $\textrm{G}$ of index $2$, then $\textrm{H}\cap\textrm{K}$ is a normal subgroup of $\textrm{G}$ of index $4$.
It is obvious that both $\textrm{H}$ and $\textrm{K}$ are normal subgrou... | It has to do with $H$ and $K$ being distinct and having index $2$: these hypotheses imply that none of them is contained in the other. So take an element $h\in H\smallsetminus K$. Then $\;G=hK\cup K$ since $K$ has index $2$. Also, since both $hK,K\subset HK$, we have $G=HK$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3056734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I get from $x - x^2 = \frac{1}{4}$ to $x =\frac{1}{2}$? I'm working on a text book problem where I need to sketch the graph of $y = 4x^2 - 4x+1$ by finding where the curve meets the $x$ axis.
To start out I set $y = 0$ then tried to isolate $x$ then,
$4x - 4x^2 = 1$
$x - x^2 = \frac{1}{4}$
From here I want to c... | $$4x-4x^2=1\implies 4x^2-4x+1=0\implies (2x-1)^2=0\implies 2x-1=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3056852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How did Einstein integrate $\frac{\partial \tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial \tau}{\partial t}=0$?
In his paper "On the Electrodynamics of Moving Bodies", Einstein writes the equation
$$\dfrac{\partial \tau}{\partial x'}+\dfrac{v}{c^2-v^2}\dfrac{\partial \tau}{\partial t}=0$$
where
*
*$\tau=\... | From the definitions given:
$$\partial_{x'}\tau = A, \quad \partial_t \tau = D$$
Also:
$$\partial_{y}\tau = B = 0, \quad \partial_z \tau = C = 0$$
From the differential equation we get:
$$A + \frac{v}{c^2 - v^2}D = 0 \\
\implies A = - \frac{v}{c^2 - v^2}D$$
Now using the definition given for $\tau$:
$$\tau = - \frac{v}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Affine transformations satisfying conditions I'm asked to find ALL affine transformations from $\mathbb{R^3}$ to itself which satisfy that the point $(-1,2,2)$ is fixed and that the lines $$\textbf{a}: y-z=y+z-2=0$$
and $$\textbf{b}: z-1=x-z=0$$ are invariant.
So, I think I've found successfully ONE affine transform... | You’ve made a good start by identifying a second fixed point of the transformation. The required family of affine transformation has two degrees of freedom remaining. To see this, consider a nonsingular affine transformation on a line: it consists of a scaling followed by a translation. For each of the two invariant li... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Understanding the solution to this polynomial estimation problem Show that there exist $K, N > 0$ such that for $x ∈\mathbb{R}$,
$$x ≥ N \implies
\frac{3x^
2 − 4x + 8}{
5x + 6}
≥ Kx.$$
Solution:
For $x ≥ 4,$ we have $4x ≤ x^
2$
so that
$$3x^
2 − 4x + 8 ≥ 3x^
2 − x^
2 = 2x^
2.$$
Similarly, for $x ≥ 1,$ we have
$$5x + 6... | Essentially the method of attack that was used in this proof is to find a quadratic function that is less than the numerator and a linear function that is greater than the denominator. Then when you divide these you get a linear function which is less than the original $\frac{3x^2-4x+8}{5x+6}$.
In layman's terms the q... | {
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If irrational numbers are uncountable, then why did I find this? I understand that irrational numbers are uncountable. I've seen the proof and it makes perfect sense. However, I came up with this (most likely false) proof that says that they're countable. Chances are, I made a mistake and the proof doesn't mean anyt... | "You can't find a solid chunk (range of numbers) that does not contain a rational number." - True, this is just the statements that the rationals are dense in the reals (and also irrationals).
"Which means that irrational numbers are just points on the number line, not lines." - True
"Which means you just need to name ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3057414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 5
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100 numbers chosen on unit interval 100 numbers are chosen from the interval [0, 1], independently of each other. What is the probability that the 2nd largest # is <1/2? My reasoning was a) all 100 chosen are less then 1/2
b) 99 are less then 1/2 and one is more
This would give (1/2)^100 OR the probability of 99 less... | I would go about it using order statistics. Let the 100 independent random variables are $X_1,\,X_2,\,\cdots,\,X_{100}$ where all are uniformly distributed in $[0,\,1]$. Arrange them as $$X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(99)}\leq X_{(100)}$$ Then you need to find $$Pr\left[X_{(99)}<\frac{1}{2}\right]=\sum_{i=99}^{... | {
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Why SVD is not unique but the Moore-Penrose pseudo inverse is unique? I feel confused about the uniqueness of the Moore-Penrose inverse generated from SVD.
For any matrix $A$, if $X$ satisfied $$AXA=A, XAX=X, (AX)^\mathrm{T}=AX, (XA)^\mathrm{T}=XA $$then $X$ is called the Moore-Penrose inverse of $A$.
If $A$ has the S... | The non-uniqueness of SVD can be characterized as follows: suppose that $A = P_0 \Sigma Q_0^T$ is one SVD of $A$. Moreover, suppose that the singular values of $A$ are $s_1$ with multiplicity $k_1$, $s_2$ with multiplicity $k_2$, and so forth, with $s_m = 0$ having multiplicity $k_m = n - r$. That is, we have
$$
\Lam... | {
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Why we can represent automorphisms in $\text{Gal}(\Bbb Q(\sqrt[4]{2},i)/\Bbb Q)$ as permutations in $S_4$ but not $S_8$? The splitting field of $x^4-2$ over $\Bbb Q$ is $G=\text{Gal}(\Bbb Q(\sqrt[4]{2},i)/\Bbb Q)$. By primitive element theorem, $K=\Bbb Q(\alpha)$ for some $\alpha$ and $[K:\Bbb Q]=8$. So I know that the... | Remember that if $F$ is the ground field, and $p(x)$ is an irreducible polynomial over $F$ of degree $n\geq 1$ without repeated roots, let $K$ be the splitting field of $p(x)$. $K/F$ is Galois. You have that the Galois group of $K/F$, say $G$, it is embedded on $S_n$, this is because $G$ permutes the roots.
In your pro... | {
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} |
Is this map chain-transitive Let $f:\mathbb R^2→\mathbb R^2$ be a continuous self-map and let $δ$ be a positive real number. A (finite or infinite) sequence $(x_{n})_{n≥0}$ is a $δ$-chain if $$d(f(x_{n}),x_{n+1})<δ$$ for all $n$.
The map $f$ is called chain-transitive if for every $x,y∈\mathbb R^2$ and every $δ>0$ ther... | Such a $g$ might not be Chain transitive.
Counter-example: $g(a, b) = (a+1, b)$. Then there does not exists a $\delta$ transitive chain between $x = (0,0)$ and $y = (1,0)$ whenever $\delta <1/2$: No matter what one choose $x_n = (a_n, b_n)$, one has $a_n \ge n/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3057862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show that $\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$ I was reading Advanced Integration Techniques, and found that$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$
The book provides one method using residue theorem and Laurent expansion. However, I wonder if there are other technique... | J.G. has the elementary method down. If you see a $1-x^2$ term inside your integrand, it might be wise to give a trig substitution a try. In this case, letting $x=\sin^2\theta$ works out beautifully.
There’s another less elementary way by utilizing the beta function and the Gamma function. Let me know if you need a pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3057924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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True or False, diagonalization problem
Let $B_c= \left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}$, $T:\mathbb{R}^4\rightarrow \mathbb{R}^4$ a linear operator such that \begin{equation} \det\left[T-I\lambda\right]_{B_{c}}=(2-\lambda)^4\qquad \text{and}\qquad T((0,0,0,1)) = (1,0,0,1)\end{equation}
Is T diagonal... | Since $T((0,0,0,1)) \neq 2 \cdot (0,0,0,1)$ we know that $(0,0,0,1)$ isn't in the eigenspace $\operatorname{Ker}(T-2I)$. The dimension of the eigenspace is therefore strictly less than the dimension of $\mathbb{R}^4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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find the Jordan basis of a matrix I'm trying to find the Jordan basis of the matrix $$A =\begin{bmatrix} 8 & 1 & 2 \\ -3 & 4 & -2\\ -3 & -1 & 3\end{bmatrix}$$ I've got the characteristic equation to be $CA(x) = (5-x)^3$ and hence the eigenvalue to be $5$. I started by finding $v_1$ such that $(A-5I)v_1=0$ and chose $v_... | The eigenspace $E_5$ has dimension $2$ (so the Jordan form of the matrix will have $2$ Jordan blocks) since
$$A-5I=\begin{bmatrix}
3&1&2\\-3&-1&-2\\-3&-1&-2\end{bmatrix}
$$
and it is defined by the single equation $\;3x+y+2z=0$.
You should attack the problem backwards: begin with choosing a vector $v_3=(x,y,z)$,
in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Prove that $ \binom nm \sum_{i=0}^{m} (-1)^i\frac{\binom mi}{n-m+1+i}=\frac{1}{1+n}$ I want to prove that
$$
\binom nm \sum_{i=0}^{m} (-1)^i\frac{\binom mi}{n-m+1+i}=\frac{1}{1+n}\\
$$
$$
\binom nm [ \frac{\binom m0}{n-m+1} +(-1)^1\frac{\binom m1}{n-m+2}+...
..+(-1)^m\frac{\binom mm}{n+1}]
=\frac{1}{1+n}\\
$$
*
*As... | Hint. Note that
$$x^{n-m}(1-x)^m=\sum_{i=0}^{m} (-1)^i \binom mi x^{n-m+i}$$
Now integrate with respect to $x$ over the interval $[0,1]$ and recall the definition of Beta function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Do all projections matrices take this form? Do all projection matrices take the form $P = A{(A^TA)}^{-1}A^T$? If so, can you help me derive it and explain it intuitively?
| $A(A^TA)^{-1}A^T$ is symmetric, but not all projection matrices are symmetric --- such as $\pmatrix{1&1\\ 0&0}$. Thus the answer is clearly no.
It is true, however, that all orthogonal (with respect to the usual inner product) projection matrices over $\mathbb R$ can be written in the form of $A(A^TA)^{-1}A^T$. By defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Product of two polytopes is a polytope Please have a look at my attempt for this problem.
Let $x = \begin{pmatrix} x_1\\ x_2 \\ \end{pmatrix}, x_1 \in P_1, x_2 \in P_2$.
I want to show that $x \in conv\{P_1 \times P_2\}$, i.e. $x$ can be represented as the convex combination of some points of $P_1 \times P_2$.
Withou... | There is a much easier way to do that.
Definition
let $v,w\in \Bbb R^n$. We define $$v\succeq w\iff v_i\ge
w_i\quad,\quad 1\le i\le n$$
According to this definition, a closed half-space can be defined as following$$\{x|a^Tx\le b\}$$when $a,x\in\Bbb R^n$ and $b\in \Bbb R$. Therefore an intersection of finite number o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Explanation of Markov transition function Here the definition of my course (in the picture below). Could someone explain me the Chapman-Kolomogorov equation ? I don't really understand what it mean. Also, I tried to make a parallel with discrete Markov chain, I don't see the link between continuous and discrete Markov ... | You can think of a continuous-time Markov process as being fully characterized by both an "embedded" discrete-time Markov chain that governs the probabilities of transitions between states (call them $Q_{ij}$), as well as some holding time parameters $\lambda_i$ that represent the average rates at which one transitions... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are uniformly equivalent metrics with the same bounded sets Holder equivalent? This is a follow-up to my question here. Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are un... | Here is a counterexample. Let $X = \mathbb{R}$ and $d_1(x,y) = \lvert y - x \rvert$. Define $d_2(x,y) = \lvert y - x \rvert$ if $\lvert y - x \rvert \le 1$ and $d_2(x,y) = \sqrt{\lvert y - x \rvert}$ if $\lvert y - x \rvert \ge 1$.
An obvious property of $d_2$ is that $\lvert y - x \rvert \le \lvert y' - x' \rvert$ imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Angular velocity as eigenvector of rotation map In physics the angular velocity $\omega$ usually is defined as follows:
For a map $B: \mathbb{R} \to SO(3)$ define $\Omega(t):=\dot{B}(t)B^t(t)$. This is a scew-symmetric matrix and hence there is a vector $\omega(t) \in \mathbb{R}^3$ satisfying $\Omega(t) x=\omega(t) \t... | No this is not true in general. Take for example
$$ B = \begin{pmatrix} 0 & -\sin\phi & -\cos\phi\\ 0 & \cos\phi & -\sin\phi\\ 1 & 0 &0 \end{pmatrix}$$
Then
$$\dot B = \dot\phi\begin{pmatrix} 0 & -\cos\phi & \sin\phi \\ 0 & -\sin\phi & -\cos\phi \\ 1 & 0 & 0\end{pmatrix}
\quad\text{and}\quad \Omega = \dot\phi\begin{pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3058917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Suppose $\sum_{n=1}^{\infty}\sqrt{{a_{n}}/{n}}$ is convergent. Prove that $\sum_{n=1}^{\infty}a_{n}$ is also convergent. Let $\{a_{n}\}$ be a decreasing sequence of non-negative real numbers. Suppose $\sum_{n=1}^{\infty}\sqrt{\frac{a_{n}}{n}}$ is convergent. Prove that $\sum_{n=1}^{\infty}a_{n}$ is also convergent.
My ... | Let $b_n=\sqrt{a_n}$. Then $\{b_n\}_{n\geq 1}$ is a decreasing sequence of positive real numbers and we want to show that
$$ \sum_{n\geq 1}\frac{b_n}{\sqrt{n}}<+\infty\quad\Longrightarrow\quad \sum_{n\geq 1}b_n^2 < +\infty.$$
By the Cauchy-Schwarz inequality disguised as Titu's lemma we have
$$ \sum_{n=N+1}^{2N}\frac{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3059124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Inverting without actual inverse? I'm reading Ash's "Basic Abstract Algebra" and been trying to understand the following:
For which I guess the problem lies when we try to take an element $hk\in HK$, it's inverse is $k^{-1}h^{-1}$ but $k^{-1}h^{-1}\in HK$ only if $k^{-1}h^{-1}=h^{-1} k^{-1}$.
My problem is the fol... | $\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$Allow me first of all to question your statement
when we try to take an element $hk\in HK$, its inverse is $k^{-1}h^{-1}$ but $k^{-1}h^{-1}\in HK$ only if $k^{-1}h^{-1}=h^{-1} k^{-1}$.
This is not true. Take for instance $G = S_{3}$, $H = \Span{(1 2)}$, $K = \Span... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3059251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that $\left|\sum_{k=0}^{n-1}\sin(2k+1)x\right|$ is bounded I'm currently studying Fourier series using a textbook which unfortunately provides only partial solutions to its explanations. To show the uniform convergence of a Fourier series I need to prove the assumption from the title above. Any help is ver... | You can calculate this sum explicitly:
\begin{align}
\sum_{k=0}^{n-1}\sin(2k+1)x &= \operatorname{Im}\left(\sum_{k=0}^{n-1}e^{(2k+1)ix}\right)\\
&= \operatorname{Im}\left(e^{ix}\sum_{k=0}^{n-1}\left(e^{2ix}\right)^{k}\right)\\
&= \operatorname{Im}\left(e^{ix}\frac{e^{2nix}-1}{e^{2ix}-1}\right)\\
&= \operatorname{Im}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3059363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Short method to evaluate $\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$? $$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$
I only know of L'hopital method but that is very long. Is there a shorter method to solve this?
| Another trick is to multiply both the numerator and denominator by
$(1+\tan(x/2))(1+\sin x)$ and use that
\begin{align}
1-\sin^2x&=\cos^2x,\\
1-\tan^2(x/2)&=\frac{\cos^2(x/2)-\sin^2(x/2)}{\cos^2(x/2)}=\frac{\cos x}{\cos^2(x/2)}.
\end{align}
Then you get after simplification
$$
\frac{1}{(1+\tan(x/2))^2(1+\sin x)\cos^2(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question on Hilbert Space $S$ is a non-empty subset of a Hilbert Space $H$, show that the set of all linear combinations of vectors in $S$ is dense in $H\Leftrightarrow S^{\bot}=\{0\}$
I have been able to show $\Rightarrow$ direction : Call the set of all linear combinations of vectors in $S$ as $S_1$ Say, $v\in S^{\... | Notice that $\left(\overline{\operatorname{span}S}\right)^\perp = S^\perp$. The Riesz projection theorem implies $$\left(\overline{\operatorname{span}S}\right)\oplus \left(\overline{\operatorname{span}S}\right)^\perp = H$$
so $$\text{linear span of } S \text{ is dense in } H \iff\overline{\operatorname{span}S} = H \iff... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $A=A^2$ is then $A^T A = A$? I know that for a matrix $A$:
If $A^TA = A$ then $A=A^2$
but is it if and only if? I mean:
is this true that "If $A=A^2$ then $A^TA = A$"?
| Since $$\det (A)^2 = \det (A)\det (A) = \det (A^2)= \det (A) \implies \det (A)\in\{0,1\}$$
So if $\det (A)= 1$ then exsist $A^{-1}$ so $A = I$ and the answer is yes.
If $\det(A)=0$ then examples show that answer is negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3059796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}$ I'm trying to calculate the integral $$\int_{x^2+y^2+z^2 \leq 1}\frac{dx\,dy\,dz}{x^2+y^2+(z-2)^2}.$$
I've tried in two methods:
Regular spherical coordinates, but this leads to really unfun integrals and logarithms with negative numbers inside them and othe... | Consider the integral $$I(a, b) = \int_{x^2 + y^2 \le a} \frac{dx \, dy\, }{x^2 + y^2 + b}$$ By changing to polar coordinates, we compute $$I(a,b) = \int_{0}^{2\pi} \int_{0}^{\sqrt{a}} \frac{r \, dr\, d\theta}{r^2 + b} = \pi \ln(a/b + 1)$$ Your desired integral is $$\int_{-1}^{1} I(1-z^2, (z-2)^2) \, dz = \pi \int_{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3059937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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System of equations has no solution
If the system of linear equations,
\begin{cases}
x &+ ay &+ z &= 3\\
x &+ 2y &+ 2z &= 6\\
x &+ 5y &+ 3z &= b\\
\end{cases}
has no solution, then:
*
*$a=-1,b=9$
*$a=-1,b \ne 9$
*$a\ne-1,b = 9$
*$a=1,b \ne 9$
I really fail to understand why the answer cannot be... | I would do it this way, using row reduction. The criterion for a non-homogeneous linear system to have solutions is that the matrix of the linear system (l.h.s.) and the augmented matrix have the same rank.
From this criterion we deduce readily that if this system has no solution, the matrix of the l.h.s. cannot have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Defining the Cosine Function from First Principles, intuitively Throughout most of my mathematics education, the cosine function has been defined formally either using its series expansion, as $\mathfrak{Re}(\exp i\theta)$, or as the unique solution to $y+y''=0$ with $y(0)=1$ and $y'(0) = 0$.
Although these definition... | Let $\alpha_n$ be the central angle of the first trangle in the $n$-th iteration.
It is an isosceles triangle with base length $\frac{\theta}n$ and leg length $1$. Hence $\alpha_n$ is given by the equation
$$\sin\frac{\alpha_n}2 = \frac{\theta}{2n}$$
or $\alpha_n = 2\arcsin\left(\frac{\theta}{2n}\right)$.
The coordinat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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Rationalizing denominator of $\frac{18}{\sqrt{162}}$. Cannot match textbook solution I am given this expression and asked to simplify by rationalizing the denominator:
$$\frac{18}{\sqrt{162}}$$
The solution is provided:
$\sqrt{2}$
I arrived at:
$$\frac{\sqrt{162}}{9}$$
Here is my thought process to arrive at this incor... | $\frac{\sqrt{162}}{9} = \frac{\sqrt{2 \cdot 9^2}}{9} = \frac{9\sqrt{2}}{9} = \sqrt{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can I conclude that $\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$ is infinite or it doesn't exists? $$\lim_{x\to0^+}\frac{x^2}{e^{-\frac{1}{x^2}}\cos(\frac{1}{x^2})^2}$$
My intuition is that the denominator goes to 0 faster and everything is non-negative, so the limit is positive infinity.
I cant ... | The limit$$\lim_{x\to0^+}\frac{x^2}{\exp\left(-\frac1{x^2}\right)}$$is indeed $+\infty$. Since $\dfrac1{\cos^2\left(\frac1{x^2}\right)}\geqslant1$ for each $x$ (when it is defined), you are right: the limit is $+\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Find the orthogonal bases of the space $ V $ and $ V^{\perp}$ In space $\mathbb R^3 $ f Find the orthogonal bases of the space $ V $ and $ V^{\perp}$ where $$V = \left\{ \vec{x} \in \mathbb R^3 : x_1 - 3x_2 + x_3 = 0 \right\} $$
On the begining, I know that may I ask you for basics of topic, but I truly have some doubt... | No, it's wrong. If you have an orthogonal basis $\mathcal{B} = \{v_1,\cdots, v_n\}$ of $V$, then for each $v_i$, it is perpendicular to other $v_j$'s but not to itself. So $v_i\notin V^\perp$ for all $i$. By the way, you can find an orthonormal basis of $V^\perp$ easily from the definition of $V$ as
$$
V = \{(1,-3,1)\}... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of $\frac{1}{7}e^{-2x^2}(1-4x^2)$ as $x\to\infty$ I calculated the derivative of $\frac{x}{7}*e^{-2x^2}$ and got $\frac{1}{7}e^{-2x^2}(1-4x^2)$ (I included it cause if I got that wrong calculating the rest is pointless)
I don't know how to find the limit of this function:
$$\frac{1}{7}e^{-2x^2}(1-4x^2)$$ I tried ... | Hint: Rewrite the expression as
$$\frac{1}{7}e^{-2x^2}\left(1-4x^2\right) = \frac{1-4x^2}{7e^{2x^2}}$$
Now, notice the growth of the numerator and denominator. Which grows more quickly?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Any good way to calculate $\frac {\alpha ^ n - 1 } {\alpha - 1} \pmod{c}$ I tried by multiplying modular inverse of denominator to the numerator and then taking modulo $c$, but there are problems when the inverse does not exist.
So is there a good way to solve this problem.
Constraints
$$ 1 \le \alpha \le 1e9 $$
$c$ i... | Set $S_0:=1$ and then recursively $S_k:=\alpha S_{k-1}+1 \pmod c$ for all $k=1,\dotsc,n-1$. The last value $S_{n-1}$ is what you seek.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3060775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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calculus and series If we suppose $\lim_{n\to+\infty} a_n =0,$ I have to examine if $\sum _{ n=1 }^{ \infty } n^{\frac {1}{1+a_n}}$ converges or diverges.
I used the monotone convergence theorem to show that the series diverges. If we suppose: $a_n=\dfrac {1}{n},$ then $$\lim_{n\to+\infty} a_n= \lim_{n\to+\infty} \fra... | Hint: It is absolutely diverging because $n^x$ is equal or greater than $1$ for $n \geq 1$ and $x \geq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof explanation of $``\exists x\in\mathbb{R}$ with $x^2=2"$ Can someone please help me break down the proof below from $(*)$ onwards. I'm lost at what is going on and where the proceeding steps are coming from. Is this a proof by contradiction? Why are we assuming $M^2\lt 2 , M^2\gt 2$, and choosing $\delta$ to be th... | Do you know the trichotomy law of inequality? It says that for any two numbers $a,b$, either $a < b$, or $a > b$, or $a=b$.
So now if you can prove that $a<b$ leads to a contradiction, and that $a > b$ leads to a contradiction, the only possibility left must be true, namely $a=b$. Apply this with $a=M^2$ and $b=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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Show that it doesn't exist any homomorphism $\phi: \mathbb{C} \to \mathbb{R}^{\mathbb{R}}$
*
*Considering the ring of all of the real functions $ \mathbb{R}^{\mathbb{R}}$. Show that it doesn't exist any homomorphism $\phi: \mathbb{C} \to \mathbb{R}^{\mathbb{R}}$
*Let $R$ be an integral domain and consider $f:... | For your other question: if $-1 \neq 1$, then $f $ can't be injective since $f(-1) = f(1) = 1$. And if $-1 = 1$, then $2=0$ and so $f$ respects addition as per your calculation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Find all 2 by 2 complex matrix with the following condition
Find all $A \in \mathrm{Mat}_{2 \times 2} ( \mathbb{C})$ such that
$A^2 = -I$ and prove that there is no 2 by 2 real diagonal matrices
$A$ with $A^2 = -I$. Deduce that for every even $n$ there are
infinitely many $n \times n$ real matrices with $A^2 = -I$.
T... | Looking very good so far. However, for the $a=\pm i, d=-a$ case, you missed the possibility $c=0$ with $b$ being anything.
For even $n$, take the real $2\times2$ matrix you've found, make $\frac n2$ copies of it all along the diagonal of an $n\times n$ matrix (whether you make them all equal, or just make sure they are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uncountable sum of vectors in a Hilbert Space I am currently reading Hilbert Spaces and confused about a thing. Say, $C=\{e_\alpha : \alpha\in\mathcal{A}\}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $\sum_{\alpha\in\mathcal{A}}e_\alpha$ well defined ? I think it should be, is there ... | The sum $\sum_{\alpha\in A} v_\alpha$ does make sense. It is defined to converge to $L \in H$ if
$$\forall \varepsilon > 0 \,\exists F_0 \subseteq A \text{ finite such that }\forall F \subseteq A \text{ finite}, F \supseteq F_0 \text{ we have} \left\|\sum_{\alpha\in F}v_\alpha - L\right\| < \varepsilon$$
However, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Proving the Fibonacci identity $(−1)^{m−k}(F_{m+k+1}F_{m−k−1}−F_{m+k}F_{m−k}) =F_{k}^2+F_{k+1}^2$ Prove that for two natural numbers $m$ and $k$, where $m>k$ the following identity holds:
$$(−1)^{m−k}(F_{m+k+1}F_{m−k−1}−F_{m+k}F_{m−k}) =F_{k}^2+F_{k+1}^2$$
Here the exercise comes with a hint:
The constant is $F^2 _m$... | Here is a proof based upon Binets formula
\begin{align*}
F_k=\frac{\varphi^k-\psi^k}{\varphi-\psi}\qquad k\geq 0\tag{1}
\end{align*}
where $\varphi=\frac{1}{2}\left(1+\sqrt{5}\right), \psi=\frac{1}{2}\left(1-\sqrt{5}\right)=-1/\varphi$.
We start with the right-hand side of OPs formula and obtain
\begin{align*}
F_k^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
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Limit of $x_n=\sum_{k=np+1}^{nq}\frac{1}{k}$ using Riemann sum I am trying to find the limit of the following sequence using Riemann sum:
$$x_n=\sum_{k=np+1}^{nq}\frac{1}{k}\qquad p,q\in\mathbb{N}\quad p<q$$
I have tried to develope the expression:
$$\frac{1}{np+1}+\frac{1}{np+2}+...+\frac{1}{nq}=\frac{1}{n}(\frac{1}{p... | We just have to calculate $\int_0^{q-p}$ instead of $\int_0^1$. Then, with
$$f(x)=\frac{1}{p+x}$$
$$\int_0^{q-p}\frac{1}{p+x}dx=[ln|p+x|]_0^{q-p}=ln|p+q-p|-ln|p|=ln\frac{q}{p}$$
And we are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $|G|=p^2$, $p$ prime and $G$ is not cyslic $G\cong \Bbb Z/p\Bbb Z\times \Bbb Z/p\Bbb Z $ Here is an incomplete proof:
If $H\le G$ is a subgroup, there are only three possibilities for its cardinal: $1,p$ or $p^2$ by Lagrange.
If $|H|=p^2$ then there exists $g\in H\subset G$ of order $p^2$ and so $G$ is cyclic.
If $|... | $H$ has cardinality $p$. The map $H\to xHx^{-1}$ defined by $h\to xhx^{-1}$ is one to one and onto. I'll leave you to check that, it is very easy. Hence $xHx^{-1}$ must have the same cardinality. As for the intersection there are only two options: it is either of size 1 or $p$. If it has size $p$ then it means that $H$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3061826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Index of summation - integer? Can the index of summation of a sum be anything other than an integer? What is the reasoning behind the answer?
| The "index" of summation, the way I'm interpreting your question, corresponds to an enumeration of the elements of the set over which you are summing. For example, if you want to add the numbers $x_{1},...,x_{n}$, then you would write
$$\sum_{k=1}^{n}x_{k}$$
You can also think of this as
$$\sum_{x\in S}x$$
where $S=\{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3062040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In $\Delta ABC$ if $(\sqrt{3}-1)a=2b$, $A=3B$, then find $C$
In $\Delta ABC$ if $(\sqrt{3}-1)a=2b$, $A=3B$, then find $C$
My Attempt
$$
b=\frac{\sqrt{3}-1}{2}a\quad\& \quad \frac{A-B}{2}=B\quad\&\quad\frac{A+B}{2}=2B\\\frac{a-b}{a+b}=\frac{\tan\frac{A-B}{2}}{\tan\frac{A+B}{2}}\implies \frac{3-\sqrt{3}}{\sqrt{3}+1}=\f... | $$2\sin B=(\sqrt3-1)\sin3B=(...)(\sin B)(3-4\sin^2B)$$
As $\sin B>0,$ $$\sin^2B=\dfrac{2-\sqrt3}4$$
$$\cos2B=1-\dfrac{2-\sqrt3}2=\cos30^\circ$$
$0<2B<360^\circ,2B=360^\circ n\pm30^\circ$ for some integer $n$
$\implies2B=30^\circ$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f \circ f = 0 $, show that transformations $f + id_x$ and $f - id_x$ are isomorphisms of $X$ I have a problem with this task:
The linear transformation $f \in L(X,X)$ has property $f \circ f = 0 $
Show that transformations $f + id_x$ and $f - id_x$ are isomorphisms of $X$ space with itself
If I need to be honest... | As all the other answers have amended my comment by actually going through the computation, I just want to add two levels of generalisations of the result which I see:
i) We only need the abelian group structure on $X$. In other words, with the same proof we get the more general result:
If $A$ is an abelian group and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Is right this chromatic polynomial for this Bridge Graph? I have the following graph:
Bridge Graph with N = 8
And I need to find its chromatic polynomial.
Based in my notes, I have reached the following result:
$$Pg(x) = \frac{((x-1)^4 + (x-1))^2}{x(x-1)} $$.
It is because I have applied this formula:
$$Pg(x) = \frac{... | Using standard techniques:
$$x^8-9 x^7+36 x^6-82 x^5+114 x^4-96 x^3+45 x^2-9 x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3062333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Theorem 5.8 Baby Rudin. Some questions This is with reference to the page number 107 of Baby Rudin.
Theorem 5.8 Let $f$ be defined on $[a,b]$.If a point $x\in(a,b)$ is a
local maximum of function $f$, and if $f'(x)$ exist, then $f'(x)=0$.
I have some questions. Thanks in advance for reading and helping out.
*
*... | Answer to question 1
The derivative of $f(x)=x^3$ vanishes at $0$ but $f$ doesn’t have a minimum nor at maximum at $0$.
Answer to question 2
Take $g(x)=\vert x \vert$, again look at zero.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does $\int_0^1 f(x)=\int_0^1 xf(x)$ imply $\int_0^x f(t)$ has a root My question is whether or not the following is true:
If $f:[0,1]\to \mathbb{R}$ is a continous function such that
$$\int_0^1 f(x)dx=\int_0^1 xf(x)dx$$
then there exist $c\in(0,1)$ such that
$$\int_0^c f(x)dx=0$$
It is quite clear that in the interval ... | Hint: Note that
$$
\int_0^1 F(x)dx =\int_0^1\left(\int_0^x f(t)dt\right)dx=\int_0^1\left(\int_t^1dx\right)f(t)dt= \int_0^1 (1-t)f(t)dt
$$ where $F(x) = \int_0^x f(t)dt$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3062662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$f \in C_{00}(\mathbb{R^p},\mathbb{C})$. $ \mapsto f_t \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ uniformly continuous? Continuing from here
Let $f_t(x):=f(x+t)$
Consider $f \mapsto f_t$ which is a linear, isometric bijection from $L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})\to L_\... | This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $\varepsilon$, there exists $\delta$ such that if $s,t\in\mathbb R^p$ satisfy $\lVert t-s\rVert\lt\delta$, then $\left\lvert f(t)-f(s)\right\rvert\lt\varepsilon$.
For any $x\in\mathbb R^p$ and $s,t$ satisfyi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3062742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Liouville's theorem for harmonic functions I was reading the proof of Liouville's theorem for harmonic functions (in $\mathbb{R}^n$) in Wikipedia, but I could not understand where do they use in that proof the assumption that $f$ is bounded.
The proof -
Taken from - https://en.m.wikipedia.org/wiki/Harmonic_function
| The boundeness of $f$ is used in the proof of the fact that, given two points $x$ and $y$, the average value of $f$ on a large disk centered at $x$ and the average value of $f$ on a large disk (with the same radius) centered at $y$ will go to the same value as the radius goes to $\infty$. When that happens, the symmetr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplifying $s=w_1-c$, where $c=\frac{1}{2}(w_1-t_1+\frac{w_2-t_2}{R})$ I have to solve this easy equation, but can't find the same answer as the correction of the exercise.
We have two equations that I want to simplify (plugging $c$ into $s$)
$$c=\dfrac{1}{2}\left(w_1-t_1+\dfrac{w_2-t_2}{R}\right)$$
$$s=w_1-c$$
I find... | You first expand a bit to simplify for the coefficient of $w_1$:
$$w_1-\frac{1}{2}\left(w_1-t_1+\frac{w_2-t_2}{R}\right) = w_1-\frac{1}{2}w_1-\frac{1}{2}\left(-t_1+\frac{w_2-t_2}{R}\right)$$
$$= \frac{1}{2}w_1-\frac{1}{2}\left(-t_1+\frac{w_2-t_2}{R}\right)$$
Factoring $\frac{1}{2}w_1$ by $-\frac{1}{2}$, the expression ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral Operator in $L^2$ I was trying to do this exercise and I'm wondering if I figured it out well:
I have $\mathcal{H} := L^2(0,1)$ and $T$ the operator with integral kernel $K(x,y) = \min\{x,y\}$, $x,y \in [0,1]$. I have to show that $T$ is compact and self-adjoint.
To show that is compact I was thinking to say t... | We have
$$
\int_{(0,1)^2} |k(x,y)|^2\ \mathsf d(x\times y) = \int_0^1\int_0^1 (x\wedge y)^2\ \mathsf dx\ \mathsf dy
\leqslant \int_0^1\int_0^1\ \mathsf dx\ \mathsf dy = 1 <\infty,
$$
so T is a Hilbert-Schmidt operator and hence is compact.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Sum of set bits in every element for a natural numbers I was thinking of a mathematical puzzle with binary representation of numbers, but could not find a convincing answer myself.
Here is the puzzle: Say for some number N, I want to find the sum of the set bits of every number from 1 to N.
For example, for 5
The answ... | $F(0) = 0.$
If $2^k \le n \lt 2^{k+1}$, then $F(n) = F(n - 2^k) + F(2^k - 1) + n - 2^k + 1$.
Since $F(2^k -1) = k\,2^{k-1}$, we have $F(n) = F(n-2^k) + k\,2^{k-1} + n - 2^k + 1$.
The recursion works because the numbers between $2^k$ and $n$ all have their highest bit set (those bits give the $n - 2^k + 1$ part of the s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Scary looking integral from a movie background I watched recently this movie: https://www.imdb.com/title/tt3149038/mediaviewer/rm261224704
and saw on the poster background, (top-left) the following integral
$$\int_{\large\frac{v}{\sqrt{t}}}^{+\infty}\frac{e^{\large-\frac{u^2}{2}}du}{(1-\tfrac{y^2}{u^2})^{3/2}}$$
I also... | For the sake of completion, here's a solution to the integral.
We'll analyze the following auxiliary integral:
\begin{align*}
\frac{1}{\sqrt{\pi}}\int_{\sqrt{a^2 +1}}^{\infty} \frac{x e^{- (bx)^2}}{(x^2 -1)^{\frac32}} \mathrm{d}x
& \overset{\color{blue}{t = b^2(x^2 -1)}}{=} \frac{be^{-b^2}}{2\sqrt{\pi}}\int_{(ab)^2}^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the main difference between pointwise and uniform convergence as defined here? I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them.
Let $E$ be a non-empty subset of $\Bbb{R}$. A sequence of functions $\{f_n\}_{n\in \Bbb{N}},$ converges... | Uniform convergence is actually $\mathcal L^\infty$ convergence, i.e.
$$
f_n \rightrightarrows f [x \in E]\!\! \iff \!\! \sup_{x \in E} \vert f_n - f\vert(x) \to 0[n \to \infty].
$$
This is strictly stronger than pointwise convergence.
Alternatively, uniform convergence implies pointwise convergence, so $f_n \to f$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to show that the given set is open? Consider the following subsets of the plane $\mathbb{R}^2$:
$$X=\{(x,y)|y=0\}\cup \{(x,y)|x>0\text{ and}\; y=1/x\}$$
How to show that $A$ and $B$ are open in $X$ under subspace topology.
Efforts:
Let's define $A=\{(x,y)|y=0\}$ and $B=\{(x,y)|x>0\text{ and}\; y=1/x\}$.
To show t... | Hint: It might be easier to show that both $A,B$ are closed in $X$, and then since $X = A \cup B$, we immediately have that $A,B$ are both open in $X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3063714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving the equation $abc=cba$ in the free group. It is known that if two words $a,b$ commute in the free group $F$, then they are powers of the same word, i.e. $a=c^r$ and $b=c^s$, where $c\in F$ and $r,s \in \mathbb Z$.
What happens if there are three words $a,b,c \in F$ such that $a b c=c b a$?
Is there a similar pr... | Another way to solve this equation is to replace the product $bc$ with a new variable, $g$ say, and replace $c^{-1}a$ another new variable, $h$ say*. Then your equation becomes
$$
\begin{align*}
abc&=cba\\
c^{-1}abc&=bcc^{-1}a\\
hg&=gh
\end{align*}
$$
Therefore, solutions over the free group $F$ are assignments $(g, h)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3063847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show that $\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\frac{\pi^2}{6}$ Wolfram Alpha shows that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\zeta(2)=\frac{\pi^2}{6}$$
I want to prove this.
Attempt:
I tried to treat this as a telescoping series:
$$\begin{align}
\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}&=\sum_{n=1}^{\infty}H_n\le... | Using summation by parts we have $$\sum_{n\leq N}\frac{H_{n}}{n\left(n+1\right)}=H_{N}\sum_{n\leq N}\frac{1}{n\left(n+1\right)}-\sum_{n\leq N-1}\frac{1}{n+1}\left(\sum_{k\leq n}\frac{1}{k\left(k+1\right)}\right).$$ Clearly $$\sum_{n\leq N}\frac{1}{n\left(n+1\right)}=\left(1-\frac{1}{N+1}\right)$$ then $$\sum_{n\leq N}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3063965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
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prove $\dim(\operatorname{range}(T)) = \dim(\operatorname{range}(\sqrt{T^*T}))$ I'm a student and I'm studying linear algebra. in Polar Decomposition we have:
for a linear operator $T$, there exist a linear isometry $S$ that:
$$ T =S\sqrt{T^*T}$$
so if $S$ is a linear transformation then it must be $\dim(\operatorn... | As @egreg pointed out, for any linear maps $S,T$, the rank of $ST$ is always less than or equal to that of $T$. To see this, note that by dimension theorem $$\dim \text{ran} L =\dim \operatorname{dom} L-\dim\ker L\le\dim \operatorname{dom} L$$ for any linear map $L$. Now, the image of $ST$ can be seen as the image of
$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing a statement is true for all positive integers regarding a complex function I am not looking for an answer, for both methods I have one. I'd simply like to check if the statement needs to be proved by induction or simply through rearrangement.
Statement:
For $f(z)=\sum_{n=0}^\infty z^{2^n}$ show that for all pos... | Personally I would have proved it by induction, since "prove this for all positive integers" or whatever often screams that, but your method is valid as well. (And probably the easier of the two, too.)
| {
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"url": "https://math.stackexchange.com/questions/3064212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A graph-coloring problem where only some of the edges should be bichromatic In a standard graph-coloring problem, it is required that all edges will be bichromatic (i.e., all edges should be connected to two vertices with different colors). What is a term, and some basic references, for a graph-coloring problem in whic... | A colouring without the condition that edges be bicromatic is called by different names such as relaxed coloring, improper coloring and defective coloring. (warning: the term defective coloring is more commonly used to mean a relaxed coloring with a restriction on maximum degree allowed in colour classes; relaxed color... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3064330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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A function with a non-zero derivative, with an inverse function that has no derivative. While studying calculus, I encountered the following statement:
"Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is continuous on said neighborhood, then $f... | The continuity condition is not necessary. It's enough that $f$ be injective on some neighborhood. This said, if your function has a sequence of jump discontinuities near $x_0$, you might have that there is no open interval $U$ around $x_0$ for which $f(U)$ is also an interval. This means that $f^{-1}$ might be defined... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3064468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
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the inverse of a sum of two symmetric for schur completion? I have a up-triangulate Jacobi matrix J which can be blocked like :
$J = \begin{bmatrix}A & B\\ 0 & C\end{bmatrix} $
both A and C are up-triangulate, we can get Hessian matrix H by:
$H = J'J =\begin{bmatrix}A' & 0\\B'&C'\end{bmatrix} \begin{bmatrix}A & B\\ 0 ... | hi @Omnomnomnom you have given me a better hint to solve this problem , I'd like to list it here
I have checked your equation it's correct except one little mistaken, it should be:
$ (C'C+B'B)^{-1} = (C'C)^{-1} - (C'C)^{-1}B'(I+B(C'C)^{-1}B')^{-1}B(C'C)^{-1} $
I think it's a good solution because B is (small rows * big... | {
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"timestamp": "2023-03-29T00:00:00",
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Burton Archimedean property proof Can anyone help me understand the following proof of Archimedean property in Burton's Elementary Number Theory book:
Theorem 1.1 Archimedean property. If $a$ and $b$ are any
positive integers, then there exists a positive integer $n$
such that $na \ge b$. P... | Eventually it means that $b-(m+1)a$ is negative, but the fact that you "know" this is jumping ahead a bit. The assumption is that $na<b$ for every positive integer $n$. That has to include $n=m+1$, so you're assuming $(m+1)a<b$ which is to say $b-(m+1)a >0.$ This leads to a contradiction and then you can conclude th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3064913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Intuitive explanation of solutions to a linear diophantine equation "Given a linear diophantine equation $ax+by=c$ with a particular solution $(x_0,y_0)$ the general solution is given by $$\biggl(x_0-\frac{b}{gcd(a,b)}t,y_0+\frac{a}{gcd(a,b)}t\biggr)$$ for all $t\in \mathbb{Z}$"
I understand the proof of this theorem p... | Let's take, as an example, the case of money. We have banknotes whose values are $a$ and $b$. We want to exchange an amount of $c$.
Finding a solution $x_0, y_0$ then corresponds to finding some number of banknotes such that we exchange exactly $c$.
But, sort of as a dufus, I could insist on exchanging more bills.
So i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
solving $\lim\limits_{x\rightarrow\infty} \frac{(x^2-1) \sqrt{x + 2}-x^2\sqrt{x+1}}{x\sqrt{x + 1}}$ To investigate the convergence of a series I have to solve the folliwing limit:
\begin{equation}
\lim\limits_{x\rightarrow\infty} \frac{(x^2-1) \sqrt{x + 2}-x^2\sqrt{x+1}}{x\sqrt{x + 1}}
\end{equation}
It should be $\fra... | Another way:
\begin{eqnarray*} \frac{(x^2-1) \sqrt{x + 2}-x^2\sqrt{x+1}}{x\sqrt{x + 1}}
& = & \frac{(x-1)(x+1) \sqrt{x + 2}-x^2\sqrt{x+1}}{x\sqrt{x + 1}} \\
& = & \frac{(x-1)\sqrt{(x+1)(x + 2)}-x^2}{x} \\
& \stackrel{x^2 = x(x-1)+x}{=} & \underbrace{\frac{x-1}{x}}_{\stackrel{x \to +\infty}{\longrightarrow}1}\underbrace... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Evaluate the limit $\lim_{n\to\infty}\log_a\left(\frac{4^nn!}{n^n}\right)$
Evaluate the limit:
$$
\lim_{n\to\infty}\log_a\left(\frac{4^nn!}{n^n}\right)\\
a>0\\
a \ne 1
$$
I've started with defining another sequence. Let:
$$
y_n = a^{x_n} = \frac{4^nn!}{n^n}
$$
Consider the fraction:
$$
\frac{y_{n+1}}{y_n} = \frac{... | Hint: Note that $$\log_a{x}=\frac{\ln(x)}{\ln(a)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
There do not exist $m\times n$ and $n\times m$ rectangular matrices $A$ and $B$ . t. $AB=I_m$ (The $m\times m$ Identity matrix)
Let $m,n\in\Bbb N$ be positive with $m>n$. Prove that $\nexists A\in M_{mn}\ \&\ B\in M_{nm},\ AB=I_m$ (The $m\times m$
Identity matrix)
I have noticed that this is not true if the matrice... | Let $B$ be a matrix with more columns $B_1,\ldots, B_n$ than rows. Then there exists a $l$ s.t. $B_l = \sum_{i \not = l} c_iB_i$, where the $c_i$s are scalars.
However, if there is a matrix $A$ such that $AB = I$ (so $A$ has $m$ rows) then for each $k$, the following must hold: $A_k \cdot B_k = 1$ and $A_k \cdot B_i = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
quotient of normalizer and centralizer is cyclic group It's known that if $G/Z(G)$ is a cyclic group, then $G$ is Abelian. Since $G/Z(G)$ is just the special case $H=G$ in the $N/C$ theorem $C_G(H)\triangleleft N_G(H)$. I wonder if the below statament is true:
If $H$ is a subgroup of $G$, if $N_G(H)/C_G(H)$ is a cyclic... | Hint: $H \cap C_G(H)=Z(H)$ and $H \unlhd N_G(H)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A finite summation of double binomial coefficients I find the following identity and have checked on Mathematica, while I have no idea how to prove it:
$$\sum_{j=0}^n(-1)^{n-j}\binom{p+j}{j}\binom{n+\beta}{n-j}=\binom{p-\beta}{n}, \quad \beta>-1, \quad p>\beta-1.$$
It seems that proof by induction does not work and bec... | We have
$$
\eqalign{
& \sum\limits_{j = 0}^n {\left( { - 1} \right)^{\,n - j} \left( \matrix{
p + j \cr
j \cr} \right)\left( \matrix{
n + \beta \cr
n - j \cr} \right)} = \cr
& = \sum\limits_{j = 0}^n {\left( { - 1} \right)^{\,n} \left( \matrix{
- p - 1 \cr
j \cr} \right)\left( \matrix{
n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Do the axioms of ordered field imply that $a\cdot 0=0$ and $0<1$? We introduce some definitions:
A structure $\mathfrak{A}=\langle A,<,+,\cdot,0,1 \rangle$ where $<$ is a linear ordering, $+$ and $\cdot$ are binary operations, and $0,1$ are constants such that all properties 1-12 are satisfied is called an ordered fie... | $a \cdot 0=0$ is a consequence of the ring axioms: $a \cdot 0= a \cdot (0+0)= a \cdot 0+ a\cdot 0$.
Assume $0 > 1$. Then $-1 = 0 + (-1) > 1 + (-1) =0$. Therefore, for any $a < b$, $(-1)a < (-1)b$, hence $0 = 1 \cdot a + (-1)a = a + (-1)a < b + (-1)a < b + (-1)b= 1 \cdot b + (-1)b = 0$, a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$. If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.
I started this way: $x^a=y^b$ $=>$ $x=y^{... | Here is an alternate method.
Note that set of primes dividing $x$ and $y$ are same. Take any prime $p$ diving $x$(and
hence $y$), and let $\alpha$ is the maximum power of $p$ in $x$ and $\beta$ is the maximum power of $p$ in $y$. Then $x^a=y^b \implies p^{\alpha a}=p^{\beta b}$, which implies $a|\beta b $ and $b| \al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Counting measure in integral. Let $\mu$ counting measure on $X$ countable set.
This is correct?
$\int_{X}|f|^pd\mu=\int_{\bigcup_{k\in\mathbb{N}} x_k} |f|^pd\mu=\sum_{k\in\mathbb{N}}\int_{x_k}|f|^pd\mu=\sum_{k\in\mathbb{N}}\int |f|^p\mathcal{X}_{x_k}d\mu=\sum_{k\in\mathbb{N}} |f(x_k)|^p\int \mathcal{X}_{x_k}=\sum_{k\in... | The equalities are in principle correct provided that $X$ is not finite.
Not completely correct is the notation that is practicized.
Every $x_k$ in the equalities should be replaced by $\{x_k\}$.
Further IMV it is a bit overdone.
If $\mu$ denotes the counting measure and $X$ is countable then you can write immediate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3065922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $a,b$ such that $\mathbb R^3 = V \oplus W$ I have some doubts with this task:
Let $a,b \in \mathbb R$. In linear space $\mathbb R^3$ we have:
$$V = span([1,1+a,-2]^T,[2,6,-2-a]^T)= span(v_1,v_2) $$ and $$W = span([0,3,-1-b]^T,[2,2+b,-2]^T) = span(w_1,w_2)$$
Find $a,b$ such that $\mathbb R^3 = V \oplus W$
... | Observe that both $V$ and $W$ can be at most two dimensional each. If both are two dimensional then their intersection cannot be just $\{0\}$. If one of them is $0$ dimensional OR both are $1$ dimensional then $\Bbb{R}^3$ cannot be their direct sum. So the only two possibilities are:
*
*$V$ is $1-$dim and W is $2-$d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to prove surjectivity of determinant It may seem obvious that the determinant of a square matrix as a map is surjective (since there is always a choice of matrix entries that yields a real number). I can't prove this statement. Any clue please?
| Hint:
Given $r\in \Bbb R $, take the diagonal matrix
$\begin{bmatrix}r&0&\cdots & 0\\
0&1&\cdots & 0\\
\vdots&\vdots&\ddots&\vdots\\0 & 0 & \cdots &1\end{bmatrix} $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Convergence of $\int_1^2 \frac{1}{\sqrt \ln x} \mathrm { d}x$ by Asymptotic Comparison Test. According to the material I have,
If $f:[a,b)\to \mathbb{R} ,b\in \mathbb{R},f\geq0 $
Then:
i) If $f$ is an infinite function of real order $\alpha\lt1$ with respect to $\frac {1}{x-b}$ $\implies$ $\int_a^bf(x)\mathrm { d}x\lt\... | An issue is that you look at $b=2$. You should consider the fact that the "problem" is at $1$, and adapt your theorem (it's easy) for $f\colon (a,b]\to\mathbb{R}$ and look at the $a=1$ end.
Rewrite, for $x>1$,
$$
\sqrt{\ln x} = \sqrt{\ln(1+(x-1))}\,.
$$
This seems silly, but now, recall that
$$
\lim_{u\to 0} \frac{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does an element of a set, that can't be in a list, make that set uncountable? We can map the infinite subsets of $\mathbb N$ to the finite subsets of $\mathbb N$
The finite subset will be a prefix, of the infinite subset (that it is paired with), that has not yet been used.
$1 \mapsto \{ \color{red}{1} ,2,3,4,5,6,7,8 ... | Even though that specific infinite subset isn't in our list, we can still have all its finite prefixes in our list of finite subsets.
For instance, say the 10th infinite subset in your list is $\{4, \ldots\}$. Then the 20th is $\{4, 5, \ldots\}$. Then the 30th is $\{4, 5, 8, \ldots\}$. And so on. None of them have to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Simplify $\frac{\sqrt{mn^3}}{a^2\sqrt{c^{-3}}} * \frac{a^{-7}n^{-2}}{\sqrt{m^2c^4}}$ to $\frac{\sqrt{mnc}}{a^9cmn}$ I need to simplify $$\frac{\sqrt{mn^3}}{a^2\sqrt{c^{-3}}} \cdot \frac{a^{-7}n^{-2}}{\sqrt{m^2c^4}}$$
The solution provided is: $\dfrac{\sqrt{mnc}}{a^9cmn}$.
I'm finding this challenging. I was able to m... | Use
$$a^{-b}=\frac{1}{a^{b}} .$$
So
\begin{align}
\frac{(n\sqrt{mn})(a^{-7}n^{-2})}{(a^2\sqrt{c^{-3}})(mc^2)}&=\frac{n\sqrt{mn}}{(a^2{c^{-3/2}})(mc^2)(a^{7}n^{2})}\\
&=\frac{(\sqrt{mn})c^{3/2}}{(mc^2)(a^{9}n)}\\
&=\frac{\sqrt{mnc}}{a^{9}cmn}\\
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
How to evaluate $\lim_{x\to 0} \frac {(\sin(2x)-2\sin(x))^4}{(3+\cos(2x)-4\cos(x))^3}$? $$\lim_{x\to 0} \frac {(\sin(2x)-2\sin(x))^4}{(3+\cos(2x)-4\cos(x))^3}$$
without L'Hôpital.
I've tried using equivalences with ${(\sin(2x)-2\sin(x))^4}$ and arrived at $-x^{12}$ but I don't know how to handle ${(3+\cos(2x)-4\cos(x)... | Hint: Note that
$$ 3+\cos(2x)-4\cos(x) = 3 + 2\cos^2(x) - 1 - 4\cos(x) = 2(\cos(x)-1)^2, $$
and that
$$ \sin(2x) - 2\sin(x) = 2\sin(x)\cos(x)-2\sin(x) = 2\sin(x)(\cos(x)-1). $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Similarity of two self-adjoint operators I am wondering whether the following is correct:
Let $A$ and $B$ be two bounded self-adjoint, positive and invertible linear operators such that $\sigma(A)=\sigma(B)$ and $AB=BA$. Can we say that $A$ is necessarily similar to $B$?
I just couldn't find an answer.
Thanks.
Math.
| Even if $\sigma(A)=\sigma(B)$, similarity sees multiplicity. For instance let
$$
A=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&2\end{bmatrix},\ \ \ B=\begin{bmatrix} 1&0&0\\ 0&2&0\\0&0&2\end{bmatrix}.
$$
Then $AB=BA$, $\sigma(A)=\sigma(B)=\{1,2\}$, but they are not similar (for instance, because they don't have the same trac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Diffeomorphism between the unit ball to itself Let $T:B\to B$ be a diffeomorphism , where $B\subset\mathbb{R}^n$ is the unit ball.
I want to show there exists $x\in B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $... | One possible diffeomorphism from the unit ball in $\mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as
$$(r, \theta) \mapsto \left(\frac{r + r^3}{2}, \theta \right),$$
or in rectangular coordinates as
$$(x, y) \mapsto \frac{1 + x^2 + y^2}{2} (x, y).$$
(The first ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Converting standard form to vertex form, parental homework help I'm trying to help my son with his homework but am having trouble feeling confident that I know what the assignment is asking for.
I've been learning (maybe relearning) about standard vs vertex form quadratic equations.
The first part of the homework stat... | This is not entirely straightforward, but we'll walk through it.
You start with an equation in standard form: $y = ax^2 + bx + c$. To convert this to 'vertex form' we must complete the square.
$$y = ax^2 + bx + c$$
$$y - c = ax^2 + bx$$
$$y - c = a(x^2 + \frac{b}{a}x)$$
$$y - c + a(\frac{b^2}{4a^2}) = a(x^2 + \frac{b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3066922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can Someone Explain the Set $\{ ∈ ℝ^ ∣ d(x,0) = 1\}$? I'm trying to understand this set. I understand most of the individual parts such as $∈$ means "an element of", $ℝ$ is all real numbers, and $d(x,0) = 1$ is the distance from $x$ to $0$, as in absolute value.
I don't understand what $x$ means in this situation thoug... | The notation $d(x,0)=1$ seems to indicate that this is a problem in a general metric space. This is not necessarily the absolute value, as there can be more abstract metrics. This set is the unit ball in $\mathbb{R}^N$ with respect to this metric.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3067038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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